if it is true that given (1,...,n) there are always as many odd-permutations as even-permutations, then we get...zero! :) right?

BTW, I have some troubles with your notation [tex]det((a_1,\cdots a_n))[/tex]. What are those [tex]a_i[/tex] ?
I assume they are vectors.
I also assume those [tex]a_{i1}[/tex] and [tex]a_{in}[/tex] are instead the vectors [tex]\mathbf{a}_{i_1}[/tex] and [tex]\mathbf{a}_{i_n}[/tex]
Did I get it right?

Finally, do you know a good source in which I can find useful exercises and tricks about manipulations with this notation?

if it is true that given (1,...,n) there are always as many odd-permutations as even-permutations, then we get...zero! :) right?

I don't think so … doesn't interchanging two indices multiply both the epsilon and the determinant by -1 ?

BTW, I have some troubles with your notation [tex]det((a_1,\cdots a_n))[/tex]. What are those [tex]a_i[/tex] ?
I assume they are vectors.
I also assume those [tex]a_{i1}[/tex] and [tex]a_{in}[/tex] are instead the vectors [tex]\mathbf{a}_{i_1}[/tex] and [tex]\mathbf{a}_{i_n}[/tex]

Yup!

(I couldn't be bothered to write it properly )

Finally, do you know a good source in which I can find useful exercises and tricks about manipulations with this notation?